LMFDB - Embedded newform 4842.2.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4842,2,Mod(1,4842)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4842, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4842.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4842 = 2 \cdot 3^{2} \cdot 269 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4842.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$38.6635646587$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.4913.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + x + 1 $ x^4 - x^3 - 6*x^2 + x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 538)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.487928$ of defining polynomial
Character	$\chi$	$=$	4842.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.48793 q^{5} +0.344151 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -1.48793 q^{5} +0.344151 q^{7} +1.00000 q^{8} -1.48793 q^{10} +0.905704 q^{11} -2.24985 q^{13} +0.344151 q^{14} +1.00000 q^{16} -2.58570 q^{17} -0.688301 q^{19} -1.48793 q^{20} +0.905704 q^{22} -1.46259 q^{23} -2.78607 q^{25} -2.24985 q^{26} +0.344151 q^{28} +4.24274 q^{29} +1.92985 q^{31} +1.00000 q^{32} -2.58570 q^{34} -0.512072 q^{35} -1.84911 q^{37} -0.688301 q^{38} -1.48793 q^{40} -1.21740 q^{41} -6.59036 q^{43} +0.905704 q^{44} -1.46259 q^{46} +6.42244 q^{47} -6.88156 q^{49} -2.78607 q^{50} -2.24985 q^{52} -2.87689 q^{53} -1.34762 q^{55} +0.344151 q^{56} +4.24274 q^{58} -11.0118 q^{59} -1.75015 q^{61} +1.92985 q^{62} +1.00000 q^{64} +3.34762 q^{65} +4.34415 q^{67} -2.58570 q^{68} -0.512072 q^{70} -5.29586 q^{71} +5.90206 q^{73} -1.84911 q^{74} -0.688301 q^{76} +0.311699 q^{77} -15.1107 q^{79} -1.48793 q^{80} -1.21740 q^{82} -15.7701 q^{83} +3.84733 q^{85} -6.59036 q^{86} +0.905704 q^{88} +1.61934 q^{89} -0.774289 q^{91} -1.46259 q^{92} +6.42244 q^{94} +1.02414 q^{95} +14.3436 q^{97} -6.88156 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} - 5 q^{5} - q^{7} + 4 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^4 - 5 * q^5 - q^7 + 4 * q^8	$ 4 q + 4 q^{2} + 4 q^{4} - 5 q^{5} - q^{7} + 4 q^{8} - 5 q^{10} - 7 q^{11} + 4 q^{13} - q^{14} + 4 q^{16} - 4 q^{17} + 2 q^{19} - 5 q^{20} - 7 q^{22} - 16 q^{23} - q^{25} + 4 q^{26} - q^{28} - q^{31} + 4 q^{32} - 4 q^{34} - 3 q^{35} - 2 q^{37} + 2 q^{38} - 5 q^{40} + q^{41} + 9 q^{43} - 7 q^{44} - 16 q^{46} - 13 q^{47} - 15 q^{49} - q^{50} + 4 q^{52} - 28 q^{53} + 13 q^{55} - q^{56} - 19 q^{59} - 20 q^{61} - q^{62} + 4 q^{64} - 5 q^{65} + 15 q^{67} - 4 q^{68} - 3 q^{70} - 15 q^{71} - 7 q^{73} - 2 q^{74} + 2 q^{76} + 6 q^{77} - 17 q^{79} - 5 q^{80} + q^{82} - 6 q^{83} - 29 q^{85} + 9 q^{86} - 7 q^{88} - 20 q^{89} - 18 q^{91} - 16 q^{92} - 13 q^{94} + 6 q^{95} + 3 q^{97} - 15 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 - 5 * q^5 - q^7 + 4 * q^8 - 5 * q^10 - 7 * q^11 + 4 * q^13 - q^14 + 4 * q^16 - 4 * q^17 + 2 * q^19 - 5 * q^20 - 7 * q^22 - 16 * q^23 - q^25 + 4 * q^26 - q^28 - q^31 + 4 * q^32 - 4 * q^34 - 3 * q^35 - 2 * q^37 + 2 * q^38 - 5 * q^40 + q^41 + 9 * q^43 - 7 * q^44 - 16 * q^46 - 13 * q^47 - 15 * q^49 - q^50 + 4 * q^52 - 28 * q^53 + 13 * q^55 - q^56 - 19 * q^59 - 20 * q^61 - q^62 + 4 * q^64 - 5 * q^65 + 15 * q^67 - 4 * q^68 - 3 * q^70 - 15 * q^71 - 7 * q^73 - 2 * q^74 + 2 * q^76 + 6 * q^77 - 17 * q^79 - 5 * q^80 + q^82 - 6 * q^83 - 29 * q^85 + 9 * q^86 - 7 * q^88 - 20 * q^89 - 18 * q^91 - 16 * q^92 - 13 * q^94 + 6 * q^95 + 3 * q^97 - 15 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.48793	−0.665422	−0.332711	−	0.943029i	$-0.607963\pi$
$5$	−1.48793	−0.665422	−0.332711	+	0.943029i	$0.607963\pi$
$6$	0	0
$7$	0.344151	0.130077	0.0650384	−	0.997883i	$-0.479283\pi$
$7$	0.344151	0.130077	0.0650384	+	0.997883i	$0.479283\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.48793	−0.470524
$11$	0.905704	0.273080	0.136540	−	0.990635i	$-0.456402\pi$
$11$	0.905704	0.273080	0.136540	+	0.990635i	$0.456402\pi$
$12$	0	0
$13$	−2.24985	−0.623997	−0.311999	−	0.950083i	$-0.600998\pi$
$13$	−2.24985	−0.623997	−0.311999	+	0.950083i	$0.600998\pi$
$14$	0.344151	0.0919782
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.58570	−0.627123	−0.313562	−	0.949568i	$-0.601522\pi$
$17$	−2.58570	−0.627123	−0.313562	+	0.949568i	$0.601522\pi$
$18$	0	0
$19$	−0.688301	−0.157907	−0.0789536	−	0.996878i	$-0.525158\pi$
$19$	−0.688301	−0.157907	−0.0789536	+	0.996878i	$0.525158\pi$
$20$	−1.48793	−0.332711
$21$	0	0
$22$	0.905704	0.193097
$23$	−1.46259	−0.304971	−0.152486	−	0.988306i	$-0.548728\pi$
$23$	−1.46259	−0.304971	−0.152486	+	0.988306i	$0.548728\pi$
$24$	0	0
$25$	−2.78607	−0.557214
$26$	−2.24985	−0.441233
$27$	0	0
$28$	0.344151	0.0650384
$29$	4.24274	0.787857	0.393929	−	0.919141i	$-0.371116\pi$
$29$	4.24274	0.787857	0.393929	+	0.919141i	$0.371116\pi$
$30$	0	0
$31$	1.92985	0.346611	0.173305	−	0.984868i	$-0.444555\pi$
$31$	1.92985	0.346611	0.173305	+	0.984868i	$0.444555\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−2.58570	−0.443443
$35$	−0.512072	−0.0865559
$36$	0	0
$37$	−1.84911	−0.303991	−0.151996	−	0.988381i	$-0.548570\pi$
$37$	−1.84911	−0.303991	−0.151996	+	0.988381i	$0.548570\pi$
$38$	−0.688301	−0.111657
$39$	0	0
$40$	−1.48793	−0.235262
$41$	−1.21740	−0.190126	−0.0950631	−	0.995471i	$-0.530305\pi$
$41$	−1.21740	−0.190126	−0.0950631	+	0.995471i	$0.530305\pi$
$42$	0	0
$43$	−6.59036	−1.00502	−0.502510	−	0.864571i	$-0.667590\pi$
$43$	−6.59036	−1.00502	−0.502510	+	0.864571i	$0.667590\pi$
$44$	0.905704	0.136540
$45$	0	0
$46$	−1.46259	−0.215647
$47$	6.42244	0.936809	0.468405	−	0.883514i	$-0.344829\pi$
$47$	6.42244	0.936809	0.468405	+	0.883514i	$0.344829\pi$
$48$	0	0
$49$	−6.88156	−0.983080
$50$	−2.78607	−0.394010
$51$	0	0
$52$	−2.24985	−0.311999
$53$	−2.87689	−0.395172	−0.197586	−	0.980286i	$-0.563310\pi$
$53$	−2.87689	−0.395172	−0.197586	+	0.980286i	$0.563310\pi$
$54$	0	0
$55$	−1.34762	−0.181713
$56$	0.344151	0.0459891
$57$	0	0
$58$	4.24274	0.557099
$59$	−11.0118	−1.43361	−0.716806	−	0.697273i	$-0.754397\pi$
$59$	−11.0118	−1.43361	−0.716806	+	0.697273i	$0.754397\pi$
$60$	0	0
$61$	−1.75015	−0.224083	−0.112042	−	0.993704i	$-0.535739\pi$
$61$	−1.75015	−0.224083	−0.112042	+	0.993704i	$0.535739\pi$
$62$	1.92985	0.245091
$63$	0	0
$64$	1.00000	0.125000
$65$	3.34762	0.415221
$66$	0	0
$67$	4.34415	0.530722	0.265361	−	0.964149i	$-0.414509\pi$
$67$	4.34415	0.530722	0.265361	+	0.964149i	$0.414509\pi$
$68$	−2.58570	−0.313562
$69$	0	0
$70$	−0.512072	−0.0612043
$71$	−5.29586	−0.628503	−0.314252	−	0.949340i	$-0.601754\pi$
$71$	−5.29586	−0.628503	−0.314252	+	0.949340i	$0.601754\pi$
$72$	0	0
$73$	5.90206	0.690784	0.345392	−	0.938459i	$-0.387746\pi$
$73$	5.90206	0.690784	0.345392	+	0.938459i	$0.387746\pi$
$74$	−1.84911	−0.214954
$75$	0	0
$76$	−0.688301	−0.0789536
$77$	0.311699	0.0355213
$78$	0	0
$79$	−15.1107	−1.70009	−0.850046	−	0.526709i	$-0.823426\pi$
$79$	−15.1107	−1.70009	−0.850046	+	0.526709i	$0.823426\pi$
$80$	−1.48793	−0.166355
$81$	0	0
$82$	−1.21740	−0.134440
$83$	−15.7701	−1.73099	−0.865495	−	0.500918i	$-0.832996\pi$
$83$	−15.7701	−1.73099	−0.865495	+	0.500918i	$0.832996\pi$
$84$	0	0
$85$	3.84733	0.417302
$86$	−6.59036	−0.710657
$87$	0	0
$88$	0.905704	0.0965483
$89$	1.61934	0.171650	0.0858250	−	0.996310i	$-0.472647\pi$
$89$	1.61934	0.171650	0.0858250	+	0.996310i	$0.472647\pi$
$90$	0	0
$91$	−0.774289	−0.0811675
$92$	−1.46259	−0.152486
$93$	0	0
$94$	6.42244	0.662424
$95$	1.02414	0.105075
$96$	0	0
$97$	14.3436	1.45637	0.728184	−	0.685381i	$-0.240364\pi$
$97$	14.3436	1.45637	0.728184	+	0.685381i	$0.240364\pi$
$98$	−6.88156	−0.695143
$99$	0	0
$100$	−2.78607	−0.278607
$101$	−2.90690	−0.289247	−0.144624	−	0.989487i	$-0.546197\pi$
$101$	−2.90690	−0.289247	−0.144624	+	0.989487i	$0.546197\pi$
$102$	0	0
$103$	−12.0866	−1.19093	−0.595464	−	0.803382i	$-0.703032\pi$
$103$	−12.0866	−1.19093	−0.595464	+	0.803382i	$0.703032\pi$
$104$	−2.24985	−0.220616
$105$	0	0
$106$	−2.87689	−0.279429
$107$	7.03712	0.680304	0.340152	−	0.940370i	$-0.389521\pi$
$107$	7.03712	0.680304	0.340152	+	0.940370i	$0.389521\pi$
$108$	0	0
$109$	−4.91926	−0.471180	−0.235590	−	0.971853i	$-0.575702\pi$
$109$	−4.91926	−0.471180	−0.235590	+	0.971853i	$0.575702\pi$
$110$	−1.34762	−0.128491
$111$	0	0
$112$	0.344151	0.0325192
$113$	−10.5108	−0.988775	−0.494387	−	0.869242i	$-0.664608\pi$
$113$	−10.5108	−0.988775	−0.494387	+	0.869242i	$0.664608\pi$
$114$	0	0
$115$	2.17623	0.202934
$116$	4.24274	0.393929
$117$	0	0
$118$	−11.0118	−1.01372
$119$	−0.889869	−0.0815742
$120$	0	0
$121$	−10.1797	−0.925427
$122$	−1.75015	−0.158451
$123$	0	0
$124$	1.92985	0.173305
$125$	11.5851	1.03620
$126$	0	0
$127$	−20.0830	−1.78207	−0.891037	−	0.453930i	$-0.850021\pi$
$127$	−20.0830	−1.78207	−0.891037	+	0.453930i	$0.850021\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	3.34762	0.293606
$131$	8.54572	0.746643	0.373321	−	0.927702i	$-0.378219\pi$
$131$	8.54572	0.746643	0.373321	+	0.927702i	$0.378219\pi$
$132$	0	0
$133$	−0.236879	−0.0205401
$134$	4.34415	0.375277
$135$	0	0
$136$	−2.58570	−0.221722
$137$	15.0625	1.28687	0.643436	−	0.765500i	$-0.277508\pi$
$137$	15.0625	1.28687	0.643436	+	0.765500i	$0.277508\pi$
$138$	0	0
$139$	−2.85030	−0.241760	−0.120880	−	0.992667i	$-0.538572\pi$
$139$	−2.85030	−0.241760	−0.120880	+	0.992667i	$0.538572\pi$
$140$	−0.512072	−0.0432780
$141$	0	0
$142$	−5.29586	−0.444419
$143$	−2.03770	−0.170401
$144$	0	0
$145$	−6.31289	−0.524257
$146$	5.90206	0.488458
$147$	0	0
$148$	−1.84911	−0.151996
$149$	5.24502	0.429689	0.214844	−	0.976648i	$-0.431076\pi$
$149$	5.24502	0.429689	0.214844	+	0.976648i	$0.431076\pi$
$150$	0	0
$151$	12.0613	0.981532	0.490766	−	0.871292i	$-0.336717\pi$
$151$	12.0613	0.981532	0.490766	+	0.871292i	$0.336717\pi$
$152$	−0.688301	−0.0558286
$153$	0	0
$154$	0.311699	0.0251174
$155$	−2.87147	−0.230642
$156$	0	0
$157$	−14.4190	−1.15076	−0.575380	−	0.817887i	$-0.695146\pi$
$157$	−14.4190	−1.15076	−0.575380	+	0.817887i	$0.695146\pi$
$158$	−15.1107	−1.20215
$159$	0	0
$160$	−1.48793	−0.117631
$161$	−0.503352	−0.0396697
$162$	0	0
$163$	20.9562	1.64142	0.820708	−	0.571347i	$-0.193579\pi$
$163$	20.9562	1.64142	0.820708	+	0.571347i	$0.193579\pi$
$164$	−1.21740	−0.0950631
$165$	0	0
$166$	−15.7701	−1.22399
$167$	2.59520	0.200823	0.100411	−	0.994946i	$-0.467984\pi$
$167$	2.59520	0.200823	0.100411	+	0.994946i	$0.467984\pi$
$168$	0	0
$169$	−7.93816	−0.610627
$170$	3.84733	0.295077
$171$	0	0
$172$	−6.59036	−0.502510
$173$	15.6629	1.19083	0.595414	−	0.803419i	$-0.296988\pi$
$173$	15.6629	1.19083	0.595414	+	0.803419i	$0.296988\pi$
$174$	0	0
$175$	−0.958828	−0.0724806
$176$	0.905704	0.0682700
$177$	0	0
$178$	1.61934	0.121375
$179$	−25.1868	−1.88255	−0.941273	−	0.337646i	$-0.890369\pi$
$179$	−25.1868	−1.88255	−0.941273	+	0.337646i	$0.890369\pi$
$180$	0	0
$181$	6.06957	0.451148	0.225574	−	0.974226i	$-0.427574\pi$
$181$	6.06957	0.451148	0.225574	+	0.974226i	$0.427574\pi$
$182$	−0.774289	−0.0573941
$183$	0	0
$184$	−1.46259	−0.107824
$185$	2.75134	0.202283
$186$	0	0
$187$	−2.34187	−0.171255
$188$	6.42244	0.468405
$189$	0	0
$190$	1.02414	0.0742992
$191$	−8.11319	−0.587050	−0.293525	−	0.955951i	$-0.594828\pi$
$191$	−8.11319	−0.587050	−0.293525	+	0.955951i	$0.594828\pi$
$192$	0	0
$193$	−12.8968	−0.928333	−0.464166	−	0.885748i	$-0.653646\pi$
$193$	−12.8968	−0.928333	−0.464166	+	0.885748i	$0.653646\pi$
$194$	14.3436	1.02981
$195$	0	0
$196$	−6.88156	−0.491540
$197$	−14.4920	−1.03251	−0.516257	−	0.856434i	$-0.672675\pi$
$197$	−14.4920	−1.03251	−0.516257	+	0.856434i	$0.672675\pi$
$198$	0	0
$199$	20.5344	1.45564	0.727822	−	0.685766i	$-0.240533\pi$
$199$	20.5344	1.45564	0.727822	+	0.685766i	$0.240533\pi$
$200$	−2.78607	−0.197005
$201$	0	0
$202$	−2.90690	−0.204529
$203$	1.46014	0.102482
$204$	0	0
$205$	1.81141	0.126514
$206$	−12.0866	−0.842113
$207$	0	0
$208$	−2.24985	−0.155999
$209$	−0.623397	−0.0431213
$210$	0	0
$211$	−6.99356	−0.481456	−0.240728	−	0.970593i	$-0.577386\pi$
$211$	−6.99356	−0.481456	−0.240728	+	0.970593i	$0.577386\pi$
$212$	−2.87689	−0.197586
$213$	0	0
$214$	7.03712	0.481047
$215$	9.80599	0.668763
$216$	0	0
$217$	0.664158	0.0450860
$218$	−4.91926	−0.333174
$219$	0	0
$220$	−1.34762	−0.0908567
$221$	5.81744	0.391323
$222$	0	0
$223$	−8.26630	−0.553552	−0.276776	−	0.960934i	$-0.589266\pi$
$223$	−8.26630	−0.553552	−0.276776	+	0.960934i	$0.589266\pi$
$224$	0.344151	0.0229945
$225$	0	0
$226$	−10.5108	−0.699169
$227$	3.27586	0.217427	0.108713	−	0.994073i	$-0.465327\pi$
$227$	3.27586	0.217427	0.108713	+	0.994073i	$0.465327\pi$
$228$	0	0
$229$	−3.04601	−0.201286	−0.100643	−	0.994923i	$-0.532090\pi$
$229$	−3.04601	−0.201286	−0.100643	+	0.994923i	$0.532090\pi$
$230$	2.17623	0.143496
$231$	0	0
$232$	4.24274	0.278550
$233$	−12.9540	−0.848644	−0.424322	−	0.905511i	$-0.639488\pi$
$233$	−12.9540	−0.848644	−0.424322	+	0.905511i	$0.639488\pi$
$234$	0	0
$235$	−9.55613	−0.623373
$236$	−11.0118	−0.716806
$237$	0	0
$238$	−0.889869	−0.0576817
$239$	−8.68652	−0.561885	−0.280942	−	0.959725i	$-0.590647\pi$
$239$	−8.68652	−0.561885	−0.280942	+	0.959725i	$0.590647\pi$
$240$	0	0
$241$	−15.8491	−1.02093	−0.510465	−	0.859898i	$-0.670527\pi$
$241$	−15.8491	−1.02093	−0.510465	+	0.859898i	$0.670527\pi$
$242$	−10.1797	−0.654376
$243$	0	0
$244$	−1.75015	−0.112042
$245$	10.2393	0.654163
$246$	0	0
$247$	1.54858	0.0985337
$248$	1.92985	0.122545
$249$	0	0
$250$	11.5851	0.732707
$251$	−4.61103	−0.291046	−0.145523	−	0.989355i	$-0.546486\pi$
$251$	−4.61103	−0.291046	−0.145523	+	0.989355i	$0.546486\pi$
$252$	0	0
$253$	−1.32467	−0.0832815
$254$	−20.0830	−1.26012
$255$	0	0
$256$	1.00000	0.0625000
$257$	−20.2182	−1.26118	−0.630588	−	0.776118i	$-0.717186\pi$
$257$	−20.2182	−1.26118	−0.630588	+	0.776118i	$0.717186\pi$
$258$	0	0
$259$	−0.636372	−0.0395422
$260$	3.34762	0.207611
$261$	0	0
$262$	8.54572	0.527956
$263$	−15.7471	−0.971009	−0.485504	−	0.874234i	$-0.661364\pi$
$263$	−15.7471	−0.971009	−0.485504	+	0.874234i	$0.661364\pi$
$264$	0	0
$265$	4.28061	0.262956
$266$	−0.236879	−0.0145240
$267$	0	0
$268$	4.34415	0.265361
$269$	−1.00000	−0.0609711
$269$	−1.00000	−0.0609711
$270$	0	0
$271$	17.3316	1.05282	0.526410	−	0.850231i	$-0.323538\pi$
$271$	17.3316	1.05282	0.526410	+	0.850231i	$0.323538\pi$
$272$	−2.58570	−0.156781
$273$	0	0
$274$	15.0625	0.909956
$275$	−2.52335	−0.152164
$276$	0	0
$277$	−23.1578	−1.39142	−0.695708	−	0.718325i	$-0.744909\pi$
$277$	−23.1578	−1.39142	−0.695708	+	0.718325i	$0.744909\pi$
$278$	−2.85030	−0.170950
$279$	0	0
$280$	−0.512072	−0.0306021
$281$	14.0106	0.835801	0.417901	−	0.908493i	$-0.362766\pi$
$281$	14.0106	0.835801	0.417901	+	0.908493i	$0.362766\pi$
$282$	0	0
$283$	26.1135	1.55229	0.776145	−	0.630555i	$-0.217173\pi$
$283$	26.1135	1.55229	0.776145	+	0.630555i	$0.217173\pi$
$284$	−5.29586	−0.314252
$285$	0	0
$286$	−2.03770	−0.120492
$287$	−0.418970	−0.0247310
$288$	0	0
$289$	−10.3142	−0.606716
$290$	−6.31289	−0.370706
$291$	0	0
$292$	5.90206	0.345392
$293$	8.14964	0.476107	0.238053	−	0.971252i	$-0.423491\pi$
$293$	8.14964	0.476107	0.238053	+	0.971252i	$0.423491\pi$
$294$	0	0
$295$	16.3847	0.953956
$296$	−1.84911	−0.107477
$297$	0	0
$298$	5.24502	0.303836
$299$	3.29062	0.190301
$300$	0	0
$301$	−2.26808	−0.130730
$302$	12.0613	0.694048
$303$	0	0
$304$	−0.688301	−0.0394768
$305$	2.60409	0.149110
$306$	0	0
$307$	0.809130	0.0461795	0.0230898	−	0.999733i	$-0.492650\pi$
$307$	0.809130	0.0461795	0.0230898	+	0.999733i	$0.492650\pi$
$308$	0.311699	0.0177607
$309$	0	0
$310$	−2.87147	−0.163089
$311$	0.444367	0.0251977	0.0125989	−	0.999921i	$-0.495990\pi$
$311$	0.444367	0.0251977	0.0125989	+	0.999921i	$0.495990\pi$
$312$	0	0
$313$	1.05085	0.0593974	0.0296987	−	0.999559i	$-0.490545\pi$
$313$	1.05085	0.0593974	0.0296987	+	0.999559i	$0.490545\pi$
$314$	−14.4190	−0.813710
$315$	0	0
$316$	−15.1107	−0.850046
$317$	−6.46784	−0.363270	−0.181635	−	0.983366i	$-0.558139\pi$
$317$	−6.46784	−0.363270	−0.181635	+	0.983366i	$0.558139\pi$
$318$	0	0
$319$	3.84266	0.215148
$320$	−1.48793	−0.0831777
$321$	0	0
$322$	−0.503352	−0.0280507
$323$	1.77974	0.0990273
$324$	0	0
$325$	6.26825	0.347700
$326$	20.9562	1.16066
$327$	0	0
$328$	−1.21740	−0.0672198
$329$	2.21029	0.121857
$330$	0	0
$331$	22.0153	1.21007	0.605034	−	0.796200i	$-0.293160\pi$
$331$	22.0153	1.21007	0.605034	+	0.796200i	$0.293160\pi$
$332$	−15.7701	−0.865495
$333$	0	0
$334$	2.59520	0.142003
$335$	−6.46379	−0.353154
$336$	0	0
$337$	14.2647	0.777047	0.388524	−	0.921439i	$-0.372985\pi$
$337$	14.2647	0.777047	0.388524	+	0.921439i	$0.372985\pi$
$338$	−7.93816	−0.431779
$339$	0	0
$340$	3.84733	0.208651
$341$	1.74787	0.0946524
$342$	0	0
$343$	−4.77735	−0.257953
$344$	−6.59036	−0.355329
$345$	0	0
$346$	15.6629	0.842043
$347$	−26.0825	−1.40018	−0.700092	−	0.714052i	$-0.746858\pi$
$347$	−26.0825	−1.40018	−0.700092	+	0.714052i	$0.746858\pi$
$348$	0	0
$349$	24.5684	1.31512	0.657559	−	0.753403i	$-0.271589\pi$
$349$	24.5684	1.31512	0.657559	+	0.753403i	$0.271589\pi$
$350$	−0.958828	−0.0512515
$351$	0	0
$352$	0.905704	0.0482742
$353$	−8.36823	−0.445396	−0.222698	−	0.974887i	$-0.571486\pi$
$353$	−8.36823	−0.445396	−0.222698	+	0.974887i	$0.571486\pi$
$354$	0	0
$355$	7.87987	0.418220
$356$	1.61934	0.0858250
$357$	0	0
$358$	−25.1868	−1.33116
$359$	2.12835	0.112330	0.0561651	−	0.998421i	$-0.482113\pi$
$359$	2.12835	0.112330	0.0561651	+	0.998421i	$0.482113\pi$
$360$	0	0
$361$	−18.5262	−0.975065
$362$	6.06957	0.319010
$363$	0	0
$364$	−0.774289	−0.0405838
$365$	−8.78184	−0.459663
$366$	0	0
$367$	9.53496	0.497721	0.248860	−	0.968539i	$-0.419944\pi$
$367$	9.53496	0.497721	0.248860	+	0.968539i	$0.419944\pi$
$368$	−1.46259	−0.0762428
$369$	0	0
$370$	2.75134	0.143035
$371$	−0.990085	−0.0514027
$372$	0	0
$373$	16.7808	0.868875	0.434437	−	0.900702i	$-0.356947\pi$
$373$	16.7808	0.868875	0.434437	+	0.900702i	$0.356947\pi$
$374$	−2.34187	−0.121095
$375$	0	0
$376$	6.42244	0.331212
$377$	−9.54555	−0.491621
$378$	0	0
$379$	−29.0341	−1.49138	−0.745690	−	0.666293i	$-0.767880\pi$
$379$	−29.0341	−1.49138	−0.745690	+	0.666293i	$0.767880\pi$
$380$	1.02414	0.0525374
$381$	0	0
$382$	−8.11319	−0.415107
$383$	−35.5798	−1.81804	−0.909022	−	0.416749i	$-0.863169\pi$
$383$	−35.5798	−1.81804	−0.909022	+	0.416749i	$0.863169\pi$
$384$	0	0
$385$	−0.463785	−0.0236367
$386$	−12.8968	−0.656430
$387$	0	0
$388$	14.3436	0.728184
$389$	37.5839	1.90558	0.952789	−	0.303634i	$-0.0982001\pi$
$389$	37.5839	1.90558	0.952789	+	0.303634i	$0.0982001\pi$
$390$	0	0
$391$	3.78181	0.191255
$392$	−6.88156	−0.347571
$393$	0	0
$394$	−14.4920	−0.730097
$395$	22.4837	1.13128
$396$	0	0
$397$	−16.8913	−0.847750	−0.423875	−	0.905721i	$-0.639330\pi$
$397$	−16.8913	−0.847750	−0.423875	+	0.905721i	$0.639330\pi$
$398$	20.5344	1.02930
$399$	0	0
$400$	−2.78607	−0.139303
$401$	−3.68585	−0.184063	−0.0920314	−	0.995756i	$-0.529336\pi$
$401$	−3.68585	−0.184063	−0.0920314	+	0.995756i	$0.529336\pi$
$402$	0	0
$403$	−4.34187	−0.216284
$404$	−2.90690	−0.144624
$405$	0	0
$406$	1.46014	0.0724656
$407$	−1.67474	−0.0830140
$408$	0	0
$409$	18.5784	0.918643	0.459322	−	0.888270i	$-0.348093\pi$
$409$	18.5784	0.918643	0.459322	+	0.888270i	$0.348093\pi$
$410$	1.81141	0.0894590
$411$	0	0
$412$	−12.0866	−0.595464
$413$	−3.78971	−0.186480
$414$	0	0
$415$	23.4647	1.15184
$416$	−2.24985	−0.110308
$417$	0	0
$418$	−0.623397	−0.0304913
$419$	−3.30890	−0.161650	−0.0808251	−	0.996728i	$-0.525756\pi$
$419$	−3.30890	−0.161650	−0.0808251	+	0.996728i	$0.525756\pi$
$420$	0	0
$421$	7.23994	0.352853	0.176427	−	0.984314i	$-0.443546\pi$
$421$	7.23994	0.352853	0.176427	+	0.984314i	$0.443546\pi$
$422$	−6.99356	−0.340441
$423$	0	0
$424$	−2.87689	−0.139714
$425$	7.20393	0.349442
$426$	0	0
$427$	−0.602314	−0.0291480
$428$	7.03712	0.340152
$429$	0	0
$430$	9.80599	0.472887
$431$	11.1645	0.537777	0.268888	−	0.963171i	$-0.413344\pi$
$431$	11.1645	0.537777	0.268888	+	0.963171i	$0.413344\pi$
$432$	0	0
$433$	28.8143	1.38473	0.692363	−	0.721549i	$-0.256570\pi$
$433$	28.8143	1.38473	0.692363	+	0.721549i	$0.256570\pi$
$434$	0.664158	0.0318806
$435$	0	0
$436$	−4.91926	−0.235590
$437$	1.00670	0.0481571
$438$	0	0
$439$	20.4897	0.977922	0.488961	−	0.872306i	$-0.337376\pi$
$439$	20.4897	0.977922	0.488961	+	0.872306i	$0.337376\pi$
$440$	−1.34762	−0.0642454
$441$	0	0
$442$	5.81744	0.276707
$443$	−9.58503	−0.455398	−0.227699	−	0.973732i	$-0.573120\pi$
$443$	−9.58503	−0.455398	−0.227699	+	0.973732i	$0.573120\pi$
$444$	0	0
$445$	−2.40947	−0.114220
$446$	−8.26630	−0.391421
$447$	0	0
$448$	0.344151	0.0162596
$449$	21.4476	1.01218	0.506088	−	0.862482i	$-0.331091\pi$
$449$	21.4476	1.01218	0.506088	+	0.862482i	$0.331091\pi$
$450$	0	0
$451$	−1.10261	−0.0519197
$452$	−10.5108	−0.494387
$453$	0	0
$454$	3.27586	0.153744
$455$	1.15209	0.0540107
$456$	0	0
$457$	10.7176	0.501350	0.250675	−	0.968071i	$-0.419347\pi$
$457$	10.7176	0.501350	0.250675	+	0.968071i	$0.419347\pi$
$458$	−3.04601	−0.142331
$459$	0	0
$460$	2.17623	0.101467
$461$	19.6658	0.915926	0.457963	−	0.888971i	$-0.348579\pi$
$461$	19.6658	0.915926	0.457963	+	0.888971i	$0.348579\pi$
$462$	0	0
$463$	22.6226	1.05136	0.525682	−	0.850681i	$-0.323810\pi$
$463$	22.6226	1.05136	0.525682	+	0.850681i	$0.323810\pi$
$464$	4.24274	0.196964
$465$	0	0
$466$	−12.9540	−0.600082
$467$	−30.0503	−1.39056	−0.695282	−	0.718737i	$-0.744720\pi$
$467$	−30.0503	−1.39056	−0.695282	+	0.718737i	$0.744720\pi$
$468$	0	0
$469$	1.49504	0.0690347
$470$	−9.55613	−0.440792
$471$	0	0
$472$	−11.0118	−0.506858
$473$	−5.96891	−0.274451
$474$	0	0
$475$	1.91766	0.0879881
$476$	−0.889869	−0.0407871
$477$	0	0
$478$	−8.68652	−0.397312
$479$	32.6251	1.49068	0.745339	−	0.666685i	$-0.232288\pi$
$479$	32.6251	1.49068	0.745339	+	0.666685i	$0.232288\pi$
$480$	0	0
$481$	4.16022	0.189690
$482$	−15.8491	−0.721907
$483$	0	0
$484$	−10.1797	−0.462714
$485$	−21.3422	−0.969099
$486$	0	0
$487$	−4.12981	−0.187139	−0.0935697	−	0.995613i	$-0.529828\pi$
$487$	−4.12981	−0.187139	−0.0935697	+	0.995613i	$0.529828\pi$
$488$	−1.75015	−0.0792254
$489$	0	0
$490$	10.2393	0.462563
$491$	35.2225	1.58957	0.794785	−	0.606891i	$-0.207584\pi$
$491$	35.2225	1.58957	0.794785	+	0.606891i	$0.207584\pi$
$492$	0	0
$493$	−10.9704	−0.494084
$494$	1.54858	0.0696738
$495$	0	0
$496$	1.92985	0.0866527
$497$	−1.82258	−0.0817537
$498$	0	0
$499$	16.3253	0.730819	0.365409	−	0.930847i	$-0.380929\pi$
$499$	16.3253	0.730819	0.365409	+	0.930847i	$0.380929\pi$
$500$	11.5851	0.518102
$501$	0	0
$502$	−4.61103	−0.205801
$503$	−39.0571	−1.74147	−0.870736	−	0.491751i	$-0.836357\pi$
$503$	−39.0571	−1.74147	−0.870736	+	0.491751i	$0.836357\pi$
$504$	0	0
$505$	4.32526	0.192471
$506$	−1.32467	−0.0588889
$507$	0	0
$508$	−20.0830	−0.891037
$509$	−8.19139	−0.363077	−0.181539	−	0.983384i	$-0.558108\pi$
$509$	−8.19139	−0.363077	−0.181539	+	0.983384i	$0.558108\pi$
$510$	0	0
$511$	2.03120	0.0898549
$512$	1.00000	0.0441942
$513$	0	0
$514$	−20.2182	−0.891786
$515$	17.9840	0.792469
$516$	0	0
$517$	5.81683	0.255824
$518$	−0.636372	−0.0279606
$519$	0	0
$520$	3.34762	0.146803
$521$	9.81960	0.430205	0.215102	−	0.976592i	$-0.430991\pi$
$521$	9.81960	0.430205	0.215102	+	0.976592i	$0.430991\pi$
$522$	0	0
$523$	4.42973	0.193698	0.0968492	−	0.995299i	$-0.469124\pi$
$523$	4.42973	0.193698	0.0968492	+	0.995299i	$0.469124\pi$
$524$	8.54572	0.373321
$525$	0	0
$526$	−15.7471	−0.686607
$527$	−4.99000	−0.217368
$528$	0	0
$529$	−20.8608	−0.906993
$530$	4.28061	0.185938
$531$	0	0
$532$	−0.236879	−0.0102700
$533$	2.73898	0.118638
$534$	0	0
$535$	−10.4707	−0.452689
$536$	4.34415	0.187639
$537$	0	0
$538$	−1.00000	−0.0431131
$539$	−6.23265	−0.268459
$540$	0	0
$541$	3.01458	0.129607	0.0648035	−	0.997898i	$-0.479358\pi$
$541$	3.01458	0.129607	0.0648035	+	0.997898i	$0.479358\pi$
$542$	17.3316	0.744457
$543$	0	0
$544$	−2.58570	−0.110861
$545$	7.31951	0.313533
$546$	0	0
$547$	35.5411	1.51963	0.759813	−	0.650141i	$-0.225290\pi$
$547$	35.5411	1.51963	0.759813	+	0.650141i	$0.225290\pi$
$548$	15.0625	0.643436
$549$	0	0
$550$	−2.52335	−0.107596
$551$	−2.92028	−0.124408
$552$	0	0
$553$	−5.20037	−0.221142
$554$	−23.1578	−0.983880
$555$	0	0
$556$	−2.85030	−0.120880
$557$	−21.5510	−0.913144	−0.456572	−	0.889687i	$-0.650923\pi$
$557$	−21.5510	−0.913144	−0.456572	+	0.889687i	$0.650923\pi$
$558$	0	0
$559$	14.8274	0.627130
$560$	−0.512072	−0.0216390
$561$	0	0
$562$	14.0106	0.591001
$563$	−30.3310	−1.27830	−0.639150	−	0.769082i	$-0.720714\pi$
$563$	−30.3310	−1.27830	−0.639150	+	0.769082i	$0.720714\pi$
$564$	0	0
$565$	15.6393	0.657952
$566$	26.1135	1.09763
$567$	0	0
$568$	−5.29586	−0.222210
$569$	−3.78038	−0.158482	−0.0792409	−	0.996855i	$-0.525250\pi$
$569$	−3.78038	−0.158482	−0.0792409	+	0.996855i	$0.525250\pi$
$570$	0	0
$571$	18.4336	0.771424	0.385712	−	0.922619i	$-0.373956\pi$
$571$	18.4336	0.771424	0.385712	+	0.922619i	$0.373956\pi$
$572$	−2.03770	−0.0852006
$573$	0	0
$574$	−0.418970	−0.0174875
$575$	4.07488	0.169934
$576$	0	0
$577$	30.3922	1.26524	0.632622	−	0.774461i	$-0.281979\pi$
$577$	30.3922	1.26524	0.632622	+	0.774461i	$0.281979\pi$
$578$	−10.3142	−0.429013
$579$	0	0
$580$	−6.31289	−0.262129
$581$	−5.42728	−0.225161
$582$	0	0
$583$	−2.60561	−0.107913
$584$	5.90206	0.244229
$585$	0	0
$586$	8.14964	0.336658
$587$	38.4248	1.58596	0.792981	−	0.609246i	$-0.208528\pi$
$587$	38.4248	1.58596	0.792981	+	0.609246i	$0.208528\pi$
$588$	0	0
$589$	−1.32832	−0.0547323
$590$	16.3847	0.674549
$591$	0	0
$592$	−1.84911	−0.0759979
$593$	−15.0010	−0.616018	−0.308009	−	0.951383i	$-0.599663\pi$
$593$	−15.0010	−0.616018	−0.308009	+	0.951383i	$0.599663\pi$
$594$	0	0
$595$	1.32406	0.0542812
$596$	5.24502	0.214844
$597$	0	0
$598$	3.29062	0.134563
$599$	8.67407	0.354413	0.177207	−	0.984174i	$-0.443294\pi$
$599$	8.67407	0.354413	0.177207	+	0.984174i	$0.443294\pi$
$600$	0	0
$601$	−20.1244	−0.820891	−0.410445	−	0.911885i	$-0.634627\pi$
$601$	−20.1244	−0.820891	−0.410445	+	0.911885i	$0.634627\pi$
$602$	−2.26808	−0.0924400
$603$	0	0
$604$	12.0613	0.490766
$605$	15.1467	0.615800
$606$	0	0
$607$	5.35730	0.217446	0.108723	−	0.994072i	$-0.465324\pi$
$607$	5.35730	0.217446	0.108723	+	0.994072i	$0.465324\pi$
$608$	−0.688301	−0.0279143
$609$	0	0
$610$	2.60409	0.105437
$611$	−14.4496	−0.584566
$612$	0	0
$613$	21.6009	0.872454	0.436227	−	0.899837i	$-0.356314\pi$
$613$	21.6009	0.872454	0.436227	+	0.899837i	$0.356314\pi$
$614$	0.809130	0.0326538
$615$	0	0
$616$	0.311699	0.0125587
$617$	36.2231	1.45829	0.729144	−	0.684360i	$-0.239918\pi$
$617$	36.2231	1.45829	0.729144	+	0.684360i	$0.239918\pi$
$618$	0	0
$619$	31.9927	1.28590	0.642948	−	0.765910i	$-0.277711\pi$
$619$	31.9927	1.28590	0.642948	+	0.765910i	$0.277711\pi$
$620$	−2.87147	−0.115321
$621$	0	0
$622$	0.444367	0.0178175
$623$	0.557298	0.0223277
$624$	0	0
$625$	−3.30747	−0.132299
$626$	1.05085	0.0420003
$627$	0	0
$628$	−14.4190	−0.575380
$629$	4.78123	0.190640
$630$	0	0
$631$	−22.3539	−0.889894	−0.444947	−	0.895557i	$-0.646778\pi$
$631$	−22.3539	−0.889894	−0.444947	+	0.895557i	$0.646778\pi$
$632$	−15.1107	−0.601073
$633$	0	0
$634$	−6.46784	−0.256871
$635$	29.8820	1.18583
$636$	0	0
$637$	15.4825	0.613439
$638$	3.84266	0.152133
$639$	0	0
$640$	−1.48793	−0.0588155
$641$	8.18573	0.323317	0.161659	−	0.986847i	$-0.448316\pi$
$641$	8.18573	0.323317	0.161659	+	0.986847i	$0.448316\pi$
$642$	0	0
$643$	17.4703	0.688961	0.344480	−	0.938793i	$-0.388055\pi$
$643$	17.4703	0.688961	0.344480	+	0.938793i	$0.388055\pi$
$644$	−0.503352	−0.0198348
$645$	0	0
$646$	1.77974	0.0700229
$647$	44.5260	1.75050	0.875249	−	0.483673i	$-0.160697\pi$
$647$	44.5260	1.75050	0.875249	+	0.483673i	$0.160697\pi$
$648$	0	0
$649$	−9.97341	−0.391491
$650$	6.26825	0.245861
$651$	0	0
$652$	20.9562	0.820708
$653$	−37.0875	−1.45134	−0.725672	−	0.688041i	$-0.758471\pi$
$653$	−37.0875	−1.45134	−0.725672	+	0.688041i	$0.758471\pi$
$654$	0	0
$655$	−12.7154	−0.496832
$656$	−1.21740	−0.0475316
$657$	0	0
$658$	2.21029	0.0861660
$659$	−15.6928	−0.611305	−0.305652	−	0.952143i	$-0.598874\pi$
$659$	−15.6928	−0.611305	−0.305652	+	0.952143i	$0.598874\pi$
$660$	0	0
$661$	4.10083	0.159504	0.0797519	−	0.996815i	$-0.474587\pi$
$661$	4.10083	0.159504	0.0797519	+	0.996815i	$0.474587\pi$
$662$	22.0153	0.855647
$663$	0	0
$664$	−15.7701	−0.611997
$665$	0.352460	0.0136678
$666$	0	0
$667$	−6.20539	−0.240274
$668$	2.59520	0.100411
$669$	0	0
$670$	−6.46379	−0.249718
$671$	−1.58511	−0.0611926
$672$	0	0
$673$	8.56587	0.330190	0.165095	−	0.986278i	$-0.447207\pi$
$673$	8.56587	0.330190	0.165095	+	0.986278i	$0.447207\pi$
$674$	14.2647	0.549456
$675$	0	0
$676$	−7.93816	−0.305314
$677$	−15.2255	−0.585165	−0.292583	−	0.956240i	$-0.594515\pi$
$677$	−15.2255	−0.585165	−0.292583	+	0.956240i	$0.594515\pi$
$678$	0	0
$679$	4.93635	0.189440
$680$	3.84733	0.147538
$681$	0	0
$682$	1.74787	0.0669294
$683$	34.5265	1.32112	0.660560	−	0.750774i	$-0.270319\pi$
$683$	34.5265	1.32112	0.660560	+	0.750774i	$0.270319\pi$
$684$	0	0
$685$	−22.4119	−0.856313
$686$	−4.77735	−0.182400
$687$	0	0
$688$	−6.59036	−0.251255
$689$	6.47259	0.246586
$690$	0	0
$691$	32.7649	1.24644	0.623218	−	0.782048i	$-0.285825\pi$
$691$	32.7649	1.24644	0.623218	+	0.782048i	$0.285825\pi$
$692$	15.6629	0.595414
$693$	0	0
$694$	−26.0825	−0.990080
$695$	4.24105	0.160872
$696$	0	0
$697$	3.14783	0.119233
$698$	24.5684	0.929929
$699$	0	0
$700$	−0.958828	−0.0362403
$701$	4.80166	0.181356	0.0906782	−	0.995880i	$-0.471097\pi$
$701$	4.80166	0.181356	0.0906782	+	0.995880i	$0.471097\pi$
$702$	0	0
$703$	1.27274	0.0480024
$704$	0.905704	0.0341350
$705$	0	0
$706$	−8.36823	−0.314943
$707$	−1.00041	−0.0376243
$708$	0	0
$709$	−34.4618	−1.29424	−0.647120	−	0.762389i	$-0.724027\pi$
$709$	−34.4618	−1.29424	−0.647120	+	0.762389i	$0.724027\pi$
$710$	7.87987	0.295726
$711$	0	0
$712$	1.61934	0.0606874
$713$	−2.82258	−0.105706
$714$	0	0
$715$	3.03195	0.113389
$716$	−25.1868	−0.941273
$717$	0	0
$718$	2.12835	0.0794295
$719$	41.1653	1.53521	0.767604	−	0.640925i	$-0.221449\pi$
$719$	41.1653	1.53521	0.767604	+	0.640925i	$0.221449\pi$
$720$	0	0
$721$	−4.15961	−0.154912
$722$	−18.5262	−0.689475
$723$	0	0
$724$	6.06957	0.225574
$725$	−11.8206	−0.439005
$726$	0	0
$727$	−50.5549	−1.87498	−0.937489	−	0.348015i	$-0.886856\pi$
$727$	−50.5549	−1.87498	−0.937489	+	0.348015i	$0.886856\pi$
$728$	−0.774289	−0.0286971
$729$	0	0
$730$	−8.78184	−0.325031
$731$	17.0407	0.630272
$732$	0	0
$733$	−29.3797	−1.08516	−0.542581	−	0.840003i	$-0.682553\pi$
$733$	−29.3797	−1.08516	−0.542581	+	0.840003i	$0.682553\pi$
$734$	9.53496	0.351942
$735$	0	0
$736$	−1.46259	−0.0539118
$737$	3.93451	0.144930
$738$	0	0
$739$	−22.4975	−0.827584	−0.413792	−	0.910371i	$-0.635796\pi$
$739$	−22.4975	−0.827584	−0.413792	+	0.910371i	$0.635796\pi$
$740$	2.75134	0.101141
$741$	0	0
$742$	−0.990085	−0.0363472
$743$	7.49123	0.274826	0.137413	−	0.990514i	$-0.456121\pi$
$743$	7.49123	0.274826	0.137413	+	0.990514i	$0.456121\pi$
$744$	0	0
$745$	−7.80421	−0.285924
$746$	16.7808	0.614387
$747$	0	0
$748$	−2.34187	−0.0856274
$749$	2.42183	0.0884917
$750$	0	0
$751$	31.3441	1.14376	0.571881	−	0.820337i	$-0.306214\pi$
$751$	31.3441	1.14376	0.571881	+	0.820337i	$0.306214\pi$
$752$	6.42244	0.234202
$753$	0	0
$754$	−9.54555	−0.347628
$755$	−17.9463	−0.653132
$756$	0	0
$757$	−44.8626	−1.63056	−0.815280	−	0.579067i	$-0.803417\pi$
$757$	−44.8626	−1.63056	−0.815280	+	0.579067i	$0.803417\pi$
$758$	−29.0341	−1.05457
$759$	0	0
$760$	1.02414	0.0371496
$761$	18.5983	0.674189	0.337094	−	0.941471i	$-0.390556\pi$
$761$	18.5983	0.674189	0.337094	+	0.941471i	$0.390556\pi$
$762$	0	0
$763$	−1.69297	−0.0612895
$764$	−8.11319	−0.293525
$765$	0	0
$766$	−35.5798	−1.28555
$767$	24.7749	0.894570
$768$	0	0
$769$	52.1929	1.88213	0.941063	−	0.338232i	$-0.109829\pi$
$769$	52.1929	1.88213	0.941063	+	0.338232i	$0.109829\pi$
$770$	−0.463785	−0.0167137
$771$	0	0
$772$	−12.8968	−0.464166
$773$	−41.7206	−1.50059	−0.750293	−	0.661105i	$-0.770088\pi$
$773$	−41.7206	−1.50059	−0.750293	+	0.661105i	$0.770088\pi$
$774$	0	0
$775$	−5.37669	−0.193136
$776$	14.3436	0.514904
$777$	0	0
$778$	37.5839	1.34745
$779$	0.837940	0.0300223
$780$	0	0
$781$	−4.79648	−0.171632
$782$	3.78181	0.135237
$783$	0	0
$784$	−6.88156	−0.245770
$785$	21.4544	0.765740
$786$	0	0
$787$	20.8879	0.744574	0.372287	−	0.928118i	$-0.378574\pi$
$787$	20.8879	0.744574	0.372287	+	0.928118i	$0.378574\pi$
$788$	−14.4920	−0.516257
$789$	0	0
$790$	22.4837	0.799934
$791$	−3.61731	−0.128617
$792$	0	0
$793$	3.93757	0.139827
$794$	−16.8913	−0.599450
$795$	0	0
$796$	20.5344	0.727822
$797$	−20.6075	−0.729955	−0.364978	−	0.931016i	$-0.618923\pi$
$797$	−20.6075	−0.729955	−0.364978	+	0.931016i	$0.618923\pi$
$798$	0	0
$799$	−16.6065	−0.587495
$800$	−2.78607	−0.0985024
$801$	0	0
$802$	−3.68585	−0.130152
$803$	5.34552	0.188639
$804$	0	0
$805$	0.748951	0.0263971
$806$	−4.34187	−0.152936
$807$	0	0
$808$	−2.90690	−0.102264
$809$	−8.27012	−0.290762	−0.145381	−	0.989376i	$-0.546441\pi$
$809$	−8.27012	−0.290762	−0.145381	+	0.989376i	$0.546441\pi$
$810$	0	0
$811$	26.6200	0.934753	0.467377	−	0.884058i	$-0.345199\pi$
$811$	26.6200	0.934753	0.467377	+	0.884058i	$0.345199\pi$
$812$	1.46014	0.0512409
$813$	0	0
$814$	−1.67474	−0.0586997
$815$	−31.1813	−1.09223
$816$	0	0
$817$	4.53616	0.158700
$818$	18.5784	0.649579
$819$	0	0
$820$	1.81141	0.0632571
$821$	−29.4026	−1.02616	−0.513078	−	0.858342i	$-0.671495\pi$
$821$	−29.4026	−1.02616	−0.513078	+	0.858342i	$0.671495\pi$
$822$	0	0
$823$	15.4716	0.539305	0.269653	−	0.962958i	$-0.413091\pi$
$823$	15.4716	0.539305	0.269653	+	0.962958i	$0.413091\pi$
$824$	−12.0866	−0.421057
$825$	0	0
$826$	−3.78971	−0.131861
$827$	24.5579	0.853963	0.426982	−	0.904260i	$-0.359577\pi$
$827$	24.5579	0.853963	0.426982	+	0.904260i	$0.359577\pi$
$828$	0	0
$829$	20.1760	0.700741	0.350371	−	0.936611i	$-0.386056\pi$
$829$	20.1760	0.700741	0.350371	+	0.936611i	$0.386056\pi$
$830$	23.4647	0.814472
$831$	0	0
$832$	−2.24985	−0.0779997
$833$	17.7936	0.616513
$834$	0	0
$835$	−3.86147	−0.133632
$836$	−0.623397	−0.0215606
$837$	0	0
$838$	−3.30890	−0.114304
$839$	39.9874	1.38052	0.690259	−	0.723562i	$-0.257496\pi$
$839$	39.9874	1.38052	0.690259	+	0.723562i	$0.257496\pi$
$840$	0	0
$841$	−10.9992	−0.379281
$842$	7.23994	0.249505
$843$	0	0
$844$	−6.99356	−0.240728
$845$	11.8114	0.406325
$846$	0	0
$847$	−3.50335	−0.120377
$848$	−2.87689	−0.0987930
$849$	0	0
$850$	7.20393	0.247093
$851$	2.70449	0.0927086
$852$	0	0
$853$	−38.2866	−1.31091	−0.655453	−	0.755236i	$-0.727522\pi$
$853$	−38.2866	−1.31091	−0.655453	+	0.755236i	$0.727522\pi$
$854$	−0.602314	−0.0206108
$855$	0	0
$856$	7.03712	0.240524
$857$	−10.4287	−0.356239	−0.178120	−	0.984009i	$-0.557001\pi$
$857$	−10.4287	−0.356239	−0.178120	+	0.984009i	$0.557001\pi$
$858$	0	0
$859$	−31.3752	−1.07051	−0.535254	−	0.844691i	$-0.679784\pi$
$859$	−31.3752	−1.07051	−0.535254	+	0.844691i	$0.679784\pi$
$860$	9.80599	0.334381
$861$	0	0
$862$	11.1645	0.380266
$863$	−22.9420	−0.780954	−0.390477	−	0.920613i	$-0.627690\pi$
$863$	−22.9420	−0.780954	−0.390477	+	0.920613i	$0.627690\pi$
$864$	0	0
$865$	−23.3053	−0.792403
$866$	28.8143	0.979150
$867$	0	0
$868$	0.664158	0.0225430
$869$	−13.6859	−0.464261
$870$	0	0
$871$	−9.77371	−0.331169
$872$	−4.91926	−0.166587
$873$	0	0
$874$	1.00670	0.0340522
$875$	3.98703	0.134786
$876$	0	0
$877$	13.8521	0.467754	0.233877	−	0.972266i	$-0.424859\pi$
$877$	13.8521	0.467754	0.233877	+	0.972266i	$0.424859\pi$
$878$	20.4897	0.691495
$879$	0	0
$880$	−1.34762	−0.0454283
$881$	−20.0782	−0.676451	−0.338225	−	0.941065i	$-0.609827\pi$
$881$	−20.0782	−0.676451	−0.338225	+	0.941065i	$0.609827\pi$
$882$	0	0
$883$	55.3481	1.86261	0.931306	−	0.364237i	$-0.118670\pi$
$883$	55.3481	1.86261	0.931306	+	0.364237i	$0.118670\pi$
$884$	5.81744	0.195662
$885$	0	0
$886$	−9.58503	−0.322015
$887$	−1.28823	−0.0432544	−0.0216272	−	0.999766i	$-0.506885\pi$
$887$	−1.28823	−0.0432544	−0.0216272	+	0.999766i	$0.506885\pi$
$888$	0	0
$889$	−6.91156	−0.231806
$890$	−2.40947	−0.0807655
$891$	0	0
$892$	−8.26630	−0.276776
$893$	−4.42058	−0.147929
$894$	0	0
$895$	37.4761	1.25269
$896$	0.344151	0.0114973
$897$	0	0
$898$	21.4476	0.715716
$899$	8.18784	0.273080
$900$	0	0
$901$	7.43877	0.247822
$902$	−1.10261	−0.0367127
$903$	0	0
$904$	−10.5108	−0.349585
$905$	−9.03109	−0.300203
$906$	0	0
$907$	11.6760	0.387695	0.193848	−	0.981032i	$-0.437903\pi$
$907$	11.6760	0.387695	0.193848	+	0.981032i	$0.437903\pi$
$908$	3.27586	0.108713
$909$	0	0
$910$	1.15209	0.0381913
$911$	−31.6479	−1.04854	−0.524271	−	0.851551i	$-0.675662\pi$
$911$	−31.6479	−1.04854	−0.524271	+	0.851551i	$0.675662\pi$
$912$	0	0
$913$	−14.2830	−0.472698
$914$	10.7176	0.354508
$915$	0	0
$916$	−3.04601	−0.100643
$917$	2.94102	0.0971209
$918$	0	0
$919$	−49.3145	−1.62674	−0.813368	−	0.581749i	$-0.802368\pi$
$919$	−49.3145	−1.62674	−0.813368	+	0.581749i	$0.802368\pi$
$920$	2.17623	0.0717482
$921$	0	0
$922$	19.6658	0.647657
$923$	11.9149	0.392184
$924$	0	0
$925$	5.15174	0.169388
$926$	22.6226	0.743426
$927$	0	0
$928$	4.24274	0.139275
$929$	25.7689	0.845450	0.422725	−	0.906258i	$-0.361074\pi$
$929$	25.7689	0.845450	0.422725	+	0.906258i	$0.361074\pi$
$930$	0	0
$931$	4.73659	0.155235
$932$	−12.9540	−0.424322
$933$	0	0
$934$	−30.0503	−0.983277
$935$	3.48454	0.113957
$936$	0	0
$937$	−6.22816	−0.203465	−0.101733	−	0.994812i	$-0.532439\pi$
$937$	−6.22816	−0.203465	−0.101733	+	0.994812i	$0.532439\pi$
$938$	1.49504	0.0488149
$939$	0	0
$940$	−9.55613	−0.311687
$941$	−15.3930	−0.501799	−0.250900	−	0.968013i	$-0.580726\pi$
$941$	−15.3930	−0.501799	−0.250900	+	0.968013i	$0.580726\pi$
$942$	0	0
$943$	1.78056	0.0579830
$944$	−11.0118	−0.358403
$945$	0	0
$946$	−5.96891	−0.194066
$947$	19.0683	0.619635	0.309818	−	0.950796i	$-0.399732\pi$
$947$	19.0683	0.619635	0.309818	+	0.950796i	$0.399732\pi$
$948$	0	0
$949$	−13.2788	−0.431047
$950$	1.91766	0.0622170
$951$	0	0
$952$	−0.889869	−0.0288408
$953$	39.7141	1.28647	0.643234	−	0.765670i	$-0.277592\pi$
$953$	39.7141	1.28647	0.643234	+	0.765670i	$0.277592\pi$
$954$	0	0
$955$	12.0718	0.390636
$956$	−8.68652	−0.280942
$957$	0	0
$958$	32.6251	1.05407
$959$	5.18376	0.167392
$960$	0	0
$961$	−27.2757	−0.879861
$962$	4.16022	0.134131
$963$	0	0
$964$	−15.8491	−0.510465
$965$	19.1895	0.617733
$966$	0	0
$967$	−54.5429	−1.75398	−0.876991	−	0.480507i	$-0.840453\pi$
$967$	−54.5429	−1.75398	−0.876991	+	0.480507i	$0.840453\pi$
$968$	−10.1797	−0.327188
$969$	0	0
$970$	−21.3422	−0.685257
$971$	20.6894	0.663955	0.331977	−	0.943287i	$-0.392284\pi$
$971$	20.6894	0.663955	0.331977	+	0.943287i	$0.392284\pi$
$972$	0	0
$973$	−0.980934	−0.0314473
$974$	−4.12981	−0.132328
$975$	0	0
$976$	−1.75015	−0.0560208
$977$	−3.49749	−0.111895	−0.0559473	−	0.998434i	$-0.517818\pi$
$977$	−3.49749	−0.111895	−0.0559473	+	0.998434i	$0.517818\pi$
$978$	0	0
$979$	1.46664	0.0468742
$980$	10.2393	0.327081
$981$	0	0
$982$	35.2225	1.12400
$983$	−16.5884	−0.529086	−0.264543	−	0.964374i	$-0.585221\pi$
$983$	−16.5884	−0.529086	−0.264543	+	0.964374i	$0.585221\pi$
$984$	0	0
$985$	21.5631	0.687057
$986$	−10.9704	−0.349370
$987$	0	0
$988$	1.54858	0.0492668
$989$	9.63900	0.306502
$990$	0	0
$991$	−27.1297	−0.861805	−0.430902	−	0.902399i	$-0.641805\pi$
$991$	−27.1297	−0.861805	−0.430902	+	0.902399i	$0.641805\pi$
$992$	1.92985	0.0612727
$993$	0	0
$994$	−1.82258	−0.0578086
$995$	−30.5537	−0.968617
$996$	0	0
$997$	56.5029	1.78946	0.894732	−	0.446603i	$-0.147366\pi$
$997$	56.5029	1.78946	0.894732	+	0.446603i	$0.147366\pi$
$998$	16.3253	0.516767
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4842.2.a.j.1.2		4
3.2	odd	2		538.2.a.c.1.4	✓	4
12.11	even	2		4304.2.a.e.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.2.a.c.1.4	✓	4	3.2	odd	2
4304.2.a.e.1.1		4	12.11	even	2
4842.2.a.j.1.2		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4842 = 2 \cdot 3^{2} \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4842.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.48793 q^{5} +0.344151 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.48793 q^{5} +0.344151 q^{7} +1.00000 q^{8} -1.48793 q^{10} +0.905704 q^{11} -2.24985 q^{13} +0.344151 q^{14} +1.00000 q^{16} -2.58570 q^{17} -0.688301 q^{19} -1.48793 q^{20} +0.905704 q^{22} -1.46259 q^{23} -2.78607 q^{25} -2.24985 q^{26} +0.344151 q^{28} +4.24274 q^{29} +1.92985 q^{31} +1.00000 q^{32} -2.58570 q^{34} -0.512072 q^{35} -1.84911 q^{37} -0.688301 q^{38} -1.48793 q^{40} -1.21740 q^{41} -6.59036 q^{43} +0.905704 q^{44} -1.46259 q^{46} +6.42244 q^{47} -6.88156 q^{49} -2.78607 q^{50} -2.24985 q^{52} -2.87689 q^{53} -1.34762 q^{55} +0.344151 q^{56} +4.24274 q^{58} -11.0118 q^{59} -1.75015 q^{61} +1.92985 q^{62} +1.00000 q^{64} +3.34762 q^{65} +4.34415 q^{67} -2.58570 q^{68} -0.512072 q^{70} -5.29586 q^{71} +5.90206 q^{73} -1.84911 q^{74} -0.688301 q^{76} +0.311699 q^{77} -15.1107 q^{79} -1.48793 q^{80} -1.21740 q^{82} -15.7701 q^{83} +3.84733 q^{85} -6.59036 q^{86} +0.905704 q^{88} +1.61934 q^{89} -0.774289 q^{91} -1.46259 q^{92} +6.42244 q^{94} +1.02414 q^{95} +14.3436 q^{97} -6.88156 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} - 5 q^{5} - q^{7} + 4 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^4 - 5 * q^5 - q^7 + 4 * q^8	\( 4 q + 4 q^{2} + 4 q^{4} - 5 q^{5} - q^{7} + 4 q^{8} - 5 q^{10} - 7 q^{11} + 4 q^{13} - q^{14} + 4 q^{16} - 4 q^{17} + 2 q^{19} - 5 q^{20} - 7 q^{22} - 16 q^{23} - q^{25} + 4 q^{26} - q^{28} - q^{31} + 4 q^{32} - 4 q^{34} - 3 q^{35} - 2 q^{37} + 2 q^{38} - 5 q^{40} + q^{41} + 9 q^{43} - 7 q^{44} - 16 q^{46} - 13 q^{47} - 15 q^{49} - q^{50} + 4 q^{52} - 28 q^{53} + 13 q^{55} - q^{56} - 19 q^{59} - 20 q^{61} - q^{62} + 4 q^{64} - 5 q^{65} + 15 q^{67} - 4 q^{68} - 3 q^{70} - 15 q^{71} - 7 q^{73} - 2 q^{74} + 2 q^{76} + 6 q^{77} - 17 q^{79} - 5 q^{80} + q^{82} - 6 q^{83} - 29 q^{85} + 9 q^{86} - 7 q^{88} - 20 q^{89} - 18 q^{91} - 16 q^{92} - 13 q^{94} + 6 q^{95} + 3 q^{97} - 15 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 - 5 * q^5 - q^7 + 4 * q^8 - 5 * q^10 - 7 * q^11 + 4 * q^13 - q^14 + 4 * q^16 - 4 * q^17 + 2 * q^19 - 5 * q^20 - 7 * q^22 - 16 * q^23 - q^25 + 4 * q^26 - q^28 - q^31 + 4 * q^32 - 4 * q^34 - 3 * q^35 - 2 * q^37 + 2 * q^38 - 5 * q^40 + q^41 + 9 * q^43 - 7 * q^44 - 16 * q^46 - 13 * q^47 - 15 * q^49 - q^50 + 4 * q^52 - 28 * q^53 + 13 * q^55 - q^56 - 19 * q^59 - 20 * q^61 - q^62 + 4 * q^64 - 5 * q^65 + 15 * q^67 - 4 * q^68 - 3 * q^70 - 15 * q^71 - 7 * q^73 - 2 * q^74 + 2 * q^76 + 6 * q^77 - 17 * q^79 - 5 * q^80 + q^82 - 6 * q^83 - 29 * q^85 + 9 * q^86 - 7 * q^88 - 20 * q^89 - 18 * q^91 - 16 * q^92 - 13 * q^94 + 6 * q^95 + 3 * q^97 - 15 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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